The (circular) helicoid is the minimal surface having a (circular) helix as its boundary. It is the only ruled minimal surface other than the plane (Catalan 1842, do Carmo 1986). For many years, the helicoid remained the only known example of a complete embedded minimal surface of finite topology with infinite curvature. However, in 1992 a second example, known as Hoffman's minimal surface and consisting of a helicoid with a hole, was discovered (Sci. News 1992). The helicoid is the only non-rotary surface which can glide along itself (Steinhaus 1999, p. 231).

The equation of a helicoid in cylindrical coordinates is


In Cartesian coordinates, it is


It can be given in parametric form by


which has an obvious generalization to the elliptic helicoid. Writing z=-cu instead of z=cv gives a cone instead of a helicoid.

The first fundamental form coefficients of the helicoid are given by


and the second fundamental form coefficients are


giving area element

 dS=sqrt(c^2+u^2)du ^ dv.

Integrating over v in [0,theta] and u in [0,r] then gives


The Gaussian curvature is given by


and the mean curvature is


making the helicoid a minimal surface. The Gaussian curvature can be given implicitly by


The helicoid can be continuously deformed into a catenoid by the transformation


where alpha=0 corresponds to a helicoid and alpha=pi/2 to a catenoid.

If a twisted curve C (i.e., one with torsion tau!=0) rotates about a fixed axis A and, at the same time, is displaced parallel to A such that the speed of displacement is always proportional to the angular velocity of rotation, then C generates a generalized helicoid.

See also

Calculus of Variations, Catenoid, Cone, Elliptic Helicoid, Generalized Helicoid, Helix, Hoffman's Minimal Surface, Hyperbolic Helicoid, Minimal Surface, Seashell

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Catalan E. "Sur les surfaces réglées dont l'aire est un minimum." J. Math. Pure Appl. 7, 203-211, Carmo, M. P. "The Helicoid." §3.5B in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 44-45, 1986.Fischer, G. (Ed.). Plate 91 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 87, 1986.Geometry Center. "The Helicoid." "Catenoid-Helicoid Deformation." "Helicoid.", A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 449 and 644, 1997.Kreyszig, E. Differential Geometry. New York: Dover, p. 88, 1991.Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.Ogawa, A. "Helicatenoid." Mathematica J. 2, 21, 1992.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, pp. 17-18, 1986.Peterson, I. "Three Bites in a Doughnut." Sci. News 127, 168, Mar. 16, 1985."Putting a Handle on a Minimal Helicoid." Sci. News 142, 276, Oct. 24, 1992.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 231-232, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 94, 1991.

Cite this as:

Weisstein, Eric W. "Helicoid." From MathWorld--A Wolfram Web Resource.

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